Question 1
1,200 words limit
(a) [20%]
A researcher estimates using a random sample of workers the following log wage
regression
log(wagei) = β0 + β1 Si+ β2 sibsi + β3 wexpi + β4 (wexpi)
2 + ui, (1.1)
where i is the subscript for individuals and goes from 1 to n, log(wagei) is the natural
logarithm of hourly wage, Si is schooling in number of years, sibsi is the number of
siblings and wexpi is the number of years of work experience. Provide details on the
interpretation of the coefficients. Explain why schooling Si could be endogenous and
describe a potential cause of this endogeneity. Explain why the ordinary least squares
estimator of model (1.1) is biased and inconsistent. More points will be given for a
formal explanation. Make sure to adapt formulas and discussion to model (1.1).
(b) [15%]
Define two different instruments that could be used for schooling Si to solve the issue
of endogeneity in model (1.1). Define the formula for the two-stage least squares
estimation for model (1.1) using the two instruments you proposed and explain how
the first and second stages’ estimations are computed. Make sure to adapt formulas
and discussion to model (1.1).
(c) [20%]
Explain what assumptions your instrumental variables must satisfy to produce a
consistent estimation of model (1.1). Explain why you think the two instruments
proposed in (1.b) are satisfying these assumptions. Show that the instrumental
variable estimation defined in (1.b) is consistent under these assumptions.
(d) [20%]
Explain how you would test for the validity of your instrumental variables. Explain also
how you would test for whether there is an endogeneity issue in your model. Provide
details on how you would perform these tests for model (1.1).
(e) [25%] The researcher decided to estimate model (1.1) using two-stage least squares
estimation and instrumenting schooling with the birth order (brthord), which is a
variable taking value 1 if the individual is a first-born child, 2 if he/she is a second-born
child, and so on. Do you think it is a valid instrument? Notice that the model is
conditional on sibs so variation in brthord is not explained by the number of siblings.
Explain in detail your answer first under the assumption that brthord is moderately
correlated with sibs and then under the assumption that the correlation between brthord
and sibs is almost 1.
Page 4 of 9

Question 2
1,200 words limit
(a) [30%]
Discuss an empirical example of a panel data model where you would use a fixed effect
estimation rather than a random effect estimation. The example must consider a
panel data of children but you are free to consider any dependent variable. You could
consider e.g. school test scores, cognitive skills, socio-emotional skills, health
outcomes, level of competitiveness, measure of social skills, average wage for the
aspired occupation, parental time investment in the child, expenditure in child private
tuition, child health expenditure, hours of physical activity per week, time spent in
school, calories intake per day, height, birth weight, BMI, days of absence in school,
etc. Write the regression equation and provide details on the dependent and
explanatory variables, on the error term and on the interpretation of the coefficients.
Explain how you would compute the fixed effect estimation for your defined model.
(b) [35%]
What does the unobserved individual effect in the model defined in (2.a) capture?
Explain the differences in the assumptions needed for the consistency of the fixed
effect estimation and of the random effect estimation for the model you discussed in
(2.a). Why is the fixed effect estimation more appropriate than the random effect
estimation in the empirical example you discussed in (2.a)? Explain how you would
perform a test to decide whether to adopt a random effect or a fixed effect estimation.
(c) [35%]
Discuss an empirical example of a panel data model where one of the explanatory
variables is endogenous because it is correlated with unobserved variables that are
relevant to explain both the dependent variable and the endogenous variable. The
example must be different from those in lectures, seminars and past exams. Explain
under which conditions the fixed effect estimation can solve such an issue of
endogeneity. Explain which type of estimation you would adopt to solve the
endogeneity issue if the conditions for the consistency of the fixed effect estimation
were not satisfied.

Question 3
1,200 words limit
(a) [20%]
Black people are much more likely to have high blood pressure and a researcher wants
to understand whether this is caused by genetic differences between black and white
people or by differences in diet between black and white people which can lead to
overweight and higher level of triglycerides in the blood. The researcher can observe
for a sample of black and white people the following variables:
– highbp, a dummy variable taking value 1 if an individual has high blood pressure
and 0 otherwise,
– black, a dummy variable taking value 1 if an individual is black and 0 if white,
– female, a dummy variable taking value 1 for women and 0 for men,
Page 5 of 9
– bmi, the body mass index which is weight in kilograms divided by height in meters
squared,
– age, age in years,
– tgresult, level of triglycerides (mg/dL) in the blood,
– tgresult2, tgresult squared.
Provide an interpretation of the results that are reported in Table 3.1 below. Write
down the models that the researcher has estimated. Explain what estimator the
researcher has used and provide details on the optimization procedure the researcher
adopted. Discuss the interpretation of the coefficients and of the tests reported in
Table 3.1. Based on these results can you conclude that the higher propensity of black
people for high blood pressure is caused by diet? Explain in detail your answer using
any test that might be useful.
Table 3.1
Prob > F = 0.0000
F( 3, 5044) = 220.53
( 3) tgresult2 = 0
( 2) tgresult = 0
( 1) bmi = 0
. test bmi tgresult tgresult2
_cons -.4018466 .0348891 -11.52 0.000 -.4702444 -.3334488
tgresult2 -5.66e-07 1.10e-07 -5.16 0.000 -7.81e-07 -3.51e-07
tgresult .0010101 .0001101 9.17 0.000 .0007942 .001226
bmi .0272994 .0013635 20.02 0.000 .0246265 .0299724
female -.0695884 .0129744 -5.36 0.000 -.0950238 -.0441531
black .110363 .0216769 5.09 0.000 .0678669 .1528591
highbp Coefficient Std. err. t P>|t| [95% conf. interval]
Total 1210.38733 5,049 .23972813 Root MSE = .45766
Adj R-squared = 0.1263
Residual 1056.46098 5,044 .209449044 R-squared = 0.1272
Model 153.92635 5 30.7852701 Prob > F = 0.0000
F(5, 5044) = 146.98
Source SS df MS Number of obs = 5,050
. reg highbp black female bmi tgresult tgresult2
_cons -.0460855 .0140881 -3.27 0.001 -.0737008 -.0184702
age .0106311 .0002606 40.80 0.000 .0101203 .011142
female -.091399 .0089789 -10.18 0.000 -.1089993 -.0737987
black .105039 .0146385 7.18 0.000 .0763447 .1337333
highbp Coefficient Std. err. t P>|t| [95% conf. interval]
Total 2525.99749 10,350 .244057728 Root MSE = .45613
Adj R-squared = 0.1475
Residual 2152.78558 10,347 .208058914 R-squared = 0.1477
Model 373.211904 3 124.403968 Prob > F = 0.0000
F(3, 10347) = 597.93
Source SS df MS Number of obs = 10,351
. reg highbp black female age
Page 6 of 9
(b) [25%]
The researcher also produced the results in Table 3.2. Write down the models that the
researcher has estimated. Explain what type of estimator the researcher has used and
provide details on the optimization procedure the researcher adopted. Discuss the
interpretation of the coefficients and the tests reported in Table 3.2. Consider a test
procedure to decide which of the two probit models considered in Table 3.2 is to be
preferred. Explain in detail your answer.
Table 3.2
(c) [30%]
Provide an explanation line by line of the Stata code that the researcher has used to
produce the results in Tables 3.3 and 3.4. Explain in detail what the reported
‘scaleprobit’ and ‘scaleprobit2’ are. Use the results in Tables 3.3. and 3.4 to provide
comments on the average marginal effect for each explanatory variable including
tgresult. Based on these results can you conclude that the higher probability of black
_cons -3.642458 .1256051 -29.00 0.000 -3.888639 -3.396276
tgresult2 -9.67e-07 3.02e-07 -3.21 0.001 -1.56e-06 -3.76e-07
tgresult .0018013 .0003208 5.62 0.000 .0011726 .00243
age .0264713 .0011995 22.07 0.000 .0241203 .0288224
bmi .0751805 .0043272 17.37 0.000 .0666995 .0836616
female -.2607344 .0391055 -6.67 0.000 -.3373798 -.1840891
black .3658577 .0643065 5.69 0.000 .2398193 .4918961
highbp Coefficient Std. err. z P>|z| [95% conf. interval]
Log likelihood = -2801.2489 Pseudo R2 = 0.1750
Prob > chi2 = 0.0000
LR chi2(6) = 1188.38
Probit regression Number of obs = 5,050
Iteration 4: log likelihood = -2801.2489
Iteration 3: log likelihood = -2801.2489
Iteration 2: log likelihood = -2801.251
Iteration 1: log likelihood = -2804.804
Iteration 0: log likelihood = -3395.4388
. probit highbp black female bmi age tgresult tgresult2
_cons -1.529171 .0427501 -35.77 0.000 -1.61296 -1.445383
age .0297 .0007895 37.62 0.000 .0281526 .0312475
female -.2743514 .0261651 -10.49 0.000 -.3256341 -.2230687
black .302711 .042485 7.13 0.000 .2194419 .3859802
highbp Coefficient Std. err. z P>|z| [95% conf. interval]
Log likelihood = -6238.8309 Pseudo R2 = 0.1152
Prob > chi2 = 0.0000
LR chi2(3) = 1623.87
Probit regression Number of obs = 10,351
Iteration 3: log likelihood = -6238.8309
Iteration 2: log likelihood = -6238.8309
Iteration 1: log likelihood = -6241.0373
Iteration 0: log likelihood = -7050.7655
. probit highbp black female age
Page 7 of 9
people to have high blood pressure is caused by diet? How would you assess the
fitness of the model in Table 3.4.
Table 3.3
scaleprobit 10,351 .3426472 .0568408 .1919711 .3989289
Variable Obs Mean Std. dev. Min Max
. sum scaleprobit
. gen scaleprobit=normalden(xbprobit)
. predict xbprobit, xb
_cons -1.529171 .0427501 -35.77 0.000 -1.61296 -1.445383
age .0297 .0007895 37.62 0.000 .0281526 .0312475
female -.2743514 .0261651 -10.49 0.000 -.3256341 -.2230687
black .302711 .042485 7.13 0.000 .2194419 .3859802
highbp Coefficient Std. err. z P>|z| [95% conf. interval]
Log likelihood = -6238.8309 Pseudo R2 = 0.1152
Prob > chi2 = 0.0000
LR chi2(3) = 1623.87
Probit regression Number of obs = 10,351
Iteration 3: log likelihood = -6238.8309
Iteration 2: log likelihood = -6238.8309
Iteration 1: log likelihood = -6241.0373
Iteration 0: log likelihood = -7050.7655
. probit highbp black female age
Page 8 of 9
Table 3.4
(d) [25%]
By using the sample and explanatory variables described above discuss how you would
estimate a model to predict the blood pressure if the dependent variable was taking
four ordered values: 0 for normal pressure, 1 for elevated blood pressure, 2 for high
blood pressure and 3 for very high blood pressure. Write the likelihood for the model
and provide an interpretation for all the parameters included in the likelihood. Make
sure to adapt formula and discussion to this specific empirical example.
Total 5,975 4,376 10,351
1 3,576 3,507 7,083
0 2,399 869 3,268
phighbp 0 1 Total
High blood pressure
. tab phighbp highbp
. gen phighbp=highbphat>0.5
(5,301 missing values generated)
. predict highbphat, p
scaleprobit2 5,050 .3139517 .0892273 .0280051 .3989423
Variable Obs Mean Std. dev. Min Max
. sum scaleprobit2
(5,301 missing values generated)
. gen scaleprobit2=normalden(xbprobit2)
(5301 missing values generated)
. predict xbprobit2, xb
_cons -3.642458 .1256051 -29.00 0.000 -3.888639 -3.396276
tgresult2 -9.67e-07 3.02e-07 -3.21 0.001 -1.56e-06 -3.76e-07
tgresult .0018013 .0003208 5.62 0.000 .0011726 .00243
age .0264713 .0011995 22.07 0.000 .0241203 .0288224
bmi .0751805 .0043272 17.37 0.000 .0666995 .0836616
female -.2607344 .0391055 -6.67 0.000 -.3373798 -.1840891
black .3658577 .0643065 5.69 0.000 .2398193 .4918961
highbp Coefficient Std. err. z P>|z| [95% conf. interval]
Log likelihood = -2801.2489 Pseudo R2 = 0.1750
Prob > chi2 = 0.0000
LR chi2(6) = 1188.38
Probit regression Number of obs = 5,050
Iteration 4: log likelihood = -2801.2489
Iteration 3: log likelihood = -2801.2489
Iteration 2: log likelihood = -2801.251
Iteration 1: log likelihood = -2804.804
Iteration 0: log likelihood = -3395.4388
. probit highbp black female bmi age tgresult tgresult2
Page 9 of 9

Question 4
1,200 words limit
(a) [25%]
For a sample of individuals involved in car accidents you observe their insurance
reimbursement for health expenditure. Each individual got a reimbursement for
health expenditure up to a maximum of £100,000. You cannot observe the individual
health expenditure, but you can observe the reimbursement. Explain what type of
model you would use to explain the individual health expenditure in pounds using, as
explanatory variables, the value of the car before the accident in pounds, the age of
the individual in years, and the number of days of hospitalization of the individual.
Write down the model and explain how you would interpret the coefficients in this
model. Explain why the ordinary least squares estimator would be biased and
inconsistent.
(b) [20%]
By considering the sample in (4.a) and dropping all observations with health
expenditure over £100,000 consider a truncated regression. Define the truncated
model by adapting any formula to the specific example in (4.a) and explain why the
maximum likelihood estimator of this truncated model is consistent but inefficient.
(c) [30%]
Write down the likelihoods for the models discussed in (4.a) and (4.b) and define each
of the parameters and variables. Make sure to adapt formulas and discussion to the
specific example.
(d) [25%]
Now assume that you observe a sample of individuals involved in car accidents whose
insurance company has a fixed excess at £200 but no maximum threshold for
reimbursements for health expenditure. This implies that all individuals with a damage
for less than £200 do not get any payment from the insurance company. For this
reason, all people with a health expenditure for less than £200 did not file any claim
and are not included in the sample. Explain what type of model you would use to
explain the amount of health expenditure in pounds using, as explanatory variables,
the value of the car before the accident in pounds, the age of the individual in years,
and the number of days of hospitalization of the individual. Write down the model and
explain how you would interpret the coefficient in such a model.

ECO00005M_Exam_Paper

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